Finally, we want to discuss our motivation for considering \(p\) -adic
spatial rotations in the first place: it grows out of Volovich's
\(p\) -adic quantum theory, in which Euclidean space is replaced by
\(p\) -adic space to define the underlying phase space [1]},
see also [2]}, [3]}, [4]}.
The idea is to realise quantum systems as unitary representations
of the symmetry group of the \(p\) -adic space, according to Noether's
theorem [5]}, and this has been realised for the displacement operations
in position and momentum, resulting in a \(p\) -adic Heisenberg-Weyl
algebra of position and momentum operators.
What has not been done is to develop a quantum theory of p-adic
angular momentum. By the same philosophy, this is identical to the
classification of all the projective unitary irreducible representations
of \(SO(3)_p\) . We get infinitely many such representations from
reducing the group modulo \(p^k\) , noting that \(SO(3)_p \mod {p}^k\)  are
finite groups, for which the irreps can be found by standard tools [6]}.
Indeed, in [7]}, [8]}, this has been done
for reduction modulo \(p\)  and \(p^2\) , for certain odd primes.
An open question is whether all irreps of \(SO(3)_p\)  arise in this way.
We suspect that this problem cannot be solved using standard
\(p\) -adic techniques based on Hensel's Lemma, but may require non-standard
\(p\) -adic analysis and Gretel's Lemma [9]}.
